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<h1>v_ldatrace
</h1>

<h2><a name="_name"></a>PURPOSE <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
<div class="box"><strong>V_LDATRACE Calculates an LDA transform to maximize trace discriminant [a,f,B,W]=(b,w,n,c)</strong></div>

<h2><a name="_synopsis"></a>SYNOPSIS <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
<div class="box"><strong>function [a,f,B,W]=v_ldatrace(b,w,n,c) </strong></div>

<h2><a name="_description"></a>DESCRIPTION <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
<div class="fragment"><pre class="comment">V_LDATRACE Calculates an LDA transform to maximize trace discriminant [a,f,B,W]=(b,w,n,c)
 If a feature vector X can come from one of several class and W and B are respectively
 the within-class and between-class covariance matrices, then the generalized Fisher discriminant
 F=trace(W\B) is a measure of how well the feature vector discriminates between the classes.
 If we choose a rectangular (tall, skinny) transformation matrix, we can define a smaller
 feature vector Y=A'*X. The aim of this routine is to choose A to maximize the Fisher
 discriminant. We assume that W is positive definite and B is positive semi-definite.
 The input argument C allows the uset to pre-specify some of the columns of A.

 Inputs:
     w[m,m] = within class covariance matrix of x
     b[m,m] = between class covariance matrix of x [default = I]
     n is the number of columns in output matrix A [default = M]
     c[m,r] specifies the first few columns of A to be predefined values [default = null)

 Outputs:
     a[m,n] is the transformation matrix: y=a'*x
     f[n,1] gives the incremental gain in f value for successive columns of A 
     B(n,n) gives the between-class covariance matrix of y
     W[n,n] gives the within-class covariance matrix of y</pre></div>

<!-- crossreference -->
<h2><a name="_cross"></a>CROSS-REFERENCE INFORMATION <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
This function calls:
<ul style="list-style-image:url(../matlabicon.gif)">
</ul>
This function is called by:
<ul style="list-style-image:url(../matlabicon.gif)">
</ul>
<!-- crossreference -->


<h2><a name="_source"></a>SOURCE CODE <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
<div class="fragment"><pre>0001 <a name="_sub0" href="#_subfunctions" class="code">function [a,f,B,W]=v_ldatrace(b,w,n,c)</a>
0002 <span class="comment">%V_LDATRACE Calculates an LDA transform to maximize trace discriminant [a,f,B,W]=(b,w,n,c)</span>
0003 <span class="comment">% If a feature vector X can come from one of several class and W and B are respectively</span>
0004 <span class="comment">% the within-class and between-class covariance matrices, then the generalized Fisher discriminant</span>
0005 <span class="comment">% F=trace(W\B) is a measure of how well the feature vector discriminates between the classes.</span>
0006 <span class="comment">% If we choose a rectangular (tall, skinny) transformation matrix, we can define a smaller</span>
0007 <span class="comment">% feature vector Y=A'*X. The aim of this routine is to choose A to maximize the Fisher</span>
0008 <span class="comment">% discriminant. We assume that W is positive definite and B is positive semi-definite.</span>
0009 <span class="comment">% The input argument C allows the uset to pre-specify some of the columns of A.</span>
0010 <span class="comment">%</span>
0011 <span class="comment">% Inputs:</span>
0012 <span class="comment">%     w[m,m] = within class covariance matrix of x</span>
0013 <span class="comment">%     b[m,m] = between class covariance matrix of x [default = I]</span>
0014 <span class="comment">%     n is the number of columns in output matrix A [default = M]</span>
0015 <span class="comment">%     c[m,r] specifies the first few columns of A to be predefined values [default = null)</span>
0016 <span class="comment">%</span>
0017 <span class="comment">% Outputs:</span>
0018 <span class="comment">%     a[m,n] is the transformation matrix: y=a'*x</span>
0019 <span class="comment">%     f[n,1] gives the incremental gain in f value for successive columns of A</span>
0020 <span class="comment">%     B(n,n) gives the between-class covariance matrix of y</span>
0021 <span class="comment">%     W[n,n] gives the within-class covariance matrix of y</span>
0022 
0023 <span class="comment">%      Copyright (C) Mike Brookes 1997</span>
0024 <span class="comment">%      Version: $Id: v_ldatrace.m 10865 2018-09-21 17:22:45Z dmb $</span>
0025 <span class="comment">%</span>
0026 <span class="comment">%   VOICEBOX is a MATLAB toolbox for speech processing.</span>
0027 <span class="comment">%   Home page: http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/voicebox.html</span>
0028 <span class="comment">%</span>
0029 <span class="comment">%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%</span>
0030 <span class="comment">%   This program is free software; you can redistribute it and/or modify</span>
0031 <span class="comment">%   it under the terms of the GNU General Public License as published by</span>
0032 <span class="comment">%   the Free Software Foundation; either version 2 of the License, or</span>
0033 <span class="comment">%   (at your option) any later version.</span>
0034 <span class="comment">%</span>
0035 <span class="comment">%   This program is distributed in the hope that it will be useful,</span>
0036 <span class="comment">%   but WITHOUT ANY WARRANTY; without even the implied warranty of</span>
0037 <span class="comment">%   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the</span>
0038 <span class="comment">%   GNU General Public License for more details.</span>
0039 <span class="comment">%</span>
0040 <span class="comment">%   You can obtain a copy of the GNU General Public License from</span>
0041 <span class="comment">%   http://www.gnu.org/copyleft/gpl.html or by writing to</span>
0042 <span class="comment">%   Free Software Foundation, Inc.,675 Mass Ave, Cambridge, MA 02139, USA.</span>
0043 <span class="comment">%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%</span>
0044 
0045 m=size(b,1);    <span class="comment">% dimension of data vectors</span>
0046 <span class="keyword">if</span> nargin&lt;4
0047     r=0;
0048     <span class="keyword">if</span> nargin&lt;3
0049         n=m;
0050         <span class="keyword">if</span> nargin&lt;2
0051             w=eye(m);
0052         <span class="keyword">end</span>
0053     <span class="keyword">end</span>
0054 <span class="keyword">else</span>
0055     r=size(c,2);    <span class="comment">% number of columns that are pre-specified</span>
0056 <span class="keyword">end</span>
0057 <span class="keyword">if</span> r
0058     <span class="keyword">if</span> n&gt;r          <span class="comment">% need to find additional vectors</span>
0059         g=chol(w);
0060         v=g\null(c'*g');
0061         [p,l,q]=svd(v'*b*v);
0062         a(:,r+1:n)=v*p(:,1:n-r);
0063         a(:,1:r)=c;
0064     <span class="keyword">else</span>
0065         a=c;        <span class="comment">% no new vectors to find</span>
0066     <span class="keyword">end</span>
0067     <span class="keyword">if</span> nargout&gt;1
0068         ari=a/triu(qr(chol(a'*w*a))); <span class="comment">% matrix a must be of full rank</span>
0069         f=diag(ari'*b*ari);
0070     <span class="keyword">end</span>
0071 <span class="keyword">else</span>
0072     [g,d]=eig(b,w,<span class="string">'qz'</span>);
0073     [ds,is]=sort(-diag(d));
0074     a=g(:,is(1:n));
0075     <span class="keyword">if</span> nargout&gt;1
0076         f=-ds(1:n);
0077     <span class="keyword">end</span>
0078 <span class="keyword">end</span>
0079 <span class="keyword">if</span> nargout &gt; 2
0080     B=a'*b*a;
0081     <span class="keyword">if</span> nargout &gt; 3
0082         W=a'*w*a;
0083     <span class="keyword">end</span>
0084     
0085 <span class="keyword">end</span></pre></div>
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